State some version of the fundamental axiom of analysis. State the alternating series test and prove it from the fundamental axiom.

In each of the following cases state whether $\sum_{n=1}^{\infty} a_{n}$ converges or diverges and prove your result. You may use any test for convergence provided you state it correctly.

(i) $a_{n}=(-1)^{n}(\log (n+1))^{-1}$.

(ii) $a_{2 n}=(2 n)^{-2}, a_{2 n-1}=-n^{-2}$.

(iii) $a_{3 n-2}=-(2 n-1)^{-1}, a_{3 n-1}=(4 n-1)^{-1}, a_{3 n}=(4 n)^{-1}$.

(iv) $a_{2^{n}+r}=(-1)^{n}\left(2^{n}+r\right)^{-1}$ for $0 \leqslant r \leqslant 2^{n}-1, n \geqslant 0$.